Problem: All of the 4th grade teachers and students from Oak went on a field trip to an archaeology museum. Tickets were $$6.00$ each for teachers and $$2.50$ each for students, and the group paid $$34.50$ in total. The next month, the same group visited a natural history museum where the tickets cost $$18.00$ each for teachers and $$12.50$ each for students, and the group paid $$148.50$ in total. Find the number of teachers and students on the field trips.
Solution: Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${6x+2.5y = 34.5}$ ${18x+12.5y = 148.5}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-3$ ${-18x-7.5y = -103.5}$ ${18x+12.5y = 148.5}$ Add the top and bottom equations together. $ 5y = 45 $ $ y = \dfrac{45}{5}$ ${y = 9}$ Now that you know ${y = 9}$ , plug it back into $ {6x+2.5y = 34.5}$ to find $x$ ${6x + 2.5}{(9)}{= 34.5}$ $6x+22.5 = 34.5$ $6x = 12$ $x = \dfrac{12}{6}$ ${x = 2}$ You can also plug ${y = 9}$ into $ {18x+12.5y = 148.5}$ and get the same answer for $x$ ${18x + 12.5}{(9)}{= 148.5}$ ${x = 2}$ There were $2$ teachers and $9$ students on the field trips.